Question: Five standard six-sided dice are rolled.  We are told there is no three-of-a-kind, but there is a pair of dice that show the same number.  These two dice are set aside, and the other three dice are re-rolled.  What is the probability that after re-rolling these three dice, at least three of the five dice show the same value?
There are a total of $6^3=216$ possible sets of dice rolls. If at least one of the re-rolled dice matches the pair we set aside, we will have at least three dice showing the same value. But we will also have three dice showing the same value if all three re-rolled dice come up the same.

Consider the first case. There are five ways for each of the three dice NOT to match the pair, so there are $5^3=125$ ways for NONE of the three dice to match the pair, so there are $216-125=91$ ways for at least one of the three dice to match the pair.

In the second case, we need all three dice to match each other. There are $6$ ways to pick which value the three dice will have.

But we have overcounted by $1;$ both of the above cases include the outcome where all five dice match. So there are $91+6-1 = 96$ ways to have at least three dice match. So, the probability is $$\frac{\text{successful outcomes}}{\text{total outcomes}}=\frac{96}{216}=\boxed{\frac{4}{9}}.$$